Chapter 15 - Section 15.9 - Change of Variables in Multiple Integrals - 15.9 Exercise - Page 1060: 14

$u=xy$ and $v=\dfrac{y}{x}$

Let us define the region $R$ as: $\dfrac{1}{x} \lt y \lt \dfrac{4}{x}, x\lt y \lt 4x$ Since, $x \gt 0$ and $y \gt 0$, then we have $1 \lt xy \lt 4$ and $1\lt \dfrac{y}{x} \lt 4$ Let us consider $u=xy$ and $v=\dfrac{y}{x}$ This gives: $1 \lt u \lt 4$ and $1\lt v \lt 4$ Hence, $u=xy$ and $v=\dfrac{y}{x}$